If your productivity does not prove it, if they can’t see it when you return to your desk, if they just will not believe you period.
Get another job.
Oh, I did 
I’ve just got one question, msmith537: why aren’t you out looking for another job in a more professional organisation? And I find it difficult to believe that anyone would hire management consultants in their 20s.
Realy?
I know dick about management consulting, but i’ve met a few people in their twenties who worked for consulting firms. One woman i knew, who was doing her Ph.D. in French history (!!), was offered a position by the Boston Group when she was in her late twenties.
Others have said it, but it bears repeating: your problem isn’t your subordinates, it’s your peers and superiors.
Look at it from the perspective of the staff members:
Which means that nobody’s giving them clear priorities. I’d bet dollars to doughnuts that every manager is telling them that their project is the Ultimate Priority For Which All Other Tasks Must Be Dropped. Even when it’s not. Probably especially when it’s not. How can you expect them to be efficient under those circumstances? Hell, how do you expect them to actually get anything done, if they work for everyone and have no single person prioritizing projects by importance and urgency? I’ll bet my own money that you loose more productivity from interrupted work and random jerking of people off one project onto another than you do from the chronic tardiness you mentioned.
So, one is never sure precisely what level of attention to detail (or what level of managerial involvement is desired, etc.) is expected of one on a day-to-day basis, then? Manager Bob is a stickler and wants every i dotted and every t crossed before he sees it - and is willing allow for the extra time it takes to do so, but Manager Jim wants to see something ASAP - even if it’s rough and needs polishing. One manager yells at you for not hopping to their project over the projects of others and another one does the passive-aggressive thing at you when you can’t get to their work immediately - who do you placate today? Bob is a big fan of the micromanaging, Jim prefers you present him with a finished product for rubber-stamping while Jill is somewhere in between - and all three get frustrated if you mix up who wants what level of involvement?
So, if I’m not a friend/lover/relative of the Big Boss, it doesn’t really matter what the hell I do, because that promotion/incentive/raise is going to someone else?
No feedback for how good a job I’ve been doing? No suggestions for how I might improve my performance to better suit the needs of the company? I get to function in a vacuum until November?
So, some of ya’ll are new, inexperienced and come with the baggage of being a newly promoted inexperienced manager (which frequently results in insecurity expressed as “making-yourself-a-giant-pain-in-the-ass” and/or “seeing-any-failure-to-kowtow-immediately-as-a-personal-insult” and the others of you get frustrated by staff members who’re used to the ones in camp A (or traumatized by the ones in camp A, or who have just had-it-up-to-here with camp A and are lashing out at anyone in a suit).
So you want me to work a whole lot harder for less compensation than you’re willing to work? Why would I do that?
So the quality of the work I produce is essentially irrelevant, providing I can have something random on paper for you by whatever the arbitrary deadline is? Why would I put forth my best efforts (and, bluntly, best efforts take longer than half-assed efforts) when I’ll get more reward from half-assing quickly?
No offense, but your office sounds almost precisely like my idea of Work Hell. I’m shocked you manage to retain any worthwhile staff members, if you want the truth. People who are efficient, organized, meticulous, dedicated and skilled generally don’t have major issues finding another job - and one where they don’t have to put up with all the nonsense you mentioned in your OP. I’d actually bet that you loose a lot of eminently suitable people before your interview process is complete - because they can see the trainwreck coming. You probably loose the majority of the remainder of competant personnel on a regular basis - I can tell you for certain I’d be shopping my resume after a week of the bullshit you described.
From the way you describe your job, it sounds like you expect your staff members to put up with an amazing amount of bullshit for little-or-no incentive and reward. It also doesn’t sound like your workplace is encouraging competance. It sounds precisely the opposite, if you want to know the truth. Sounds like you work somewhere that encourages and fosters incompetance and discourages competance - why are you at all surprised your staff members aren’t the greatest?
Its actually really common - they get hired straight out of B-School to learn to do things “The Accenture Way” - they work their butts off (or not) for a few years and discover if they have what it takes to climb the partnership ladder (or at least midmanagement - where the pay is decent and the hours suck less), if they don’t make it, they quit and are replaced by new 24 year olds with fresh minted MBAs and they go work for Fortune 500s that were formerly their clients.
I think all the real suggestions have been covered already, so my vote is for “start drinking.” Drink early, drink often. 
But seriously, your company is getting the employees it deserves. All the good people have and will find jobs at better managed companies. This article delineates the 10 biggest causes of workplace stress, and your company is doing most of them. Interestingly enough, the article also says that the 10 biggest stressors are all caused by leadership problems. You can agitate for change from within, but that sounds like a fairly fruitless effort. I’d start shopping a resume around.
Another point my husband has learned from the construction safety industry - you would think people would buy into safety, because the bottom line is them not getting hurt or killed, but they don’t buy in if the consequences (good and bad) aren’t visible and immediate. Nobody does. Your workers also need visible and immediate feedback on their actions.
Pretty accurate, actually - there are differences betwen “Strat Houses” like the Boston Consulting Group (BCG), McKinsey, Booz Allen & Hamilton, etc. and “Solutions Integration” firms like Accenture, Deloitte, EDS, etc. As a Strat guy within a Solutions firm, I was in a schizophrenic situation - pulled in multiple directions depending on how the org was sliced at the time…
And yeah, you really can’t make it in consulting unless you are prepared to work like a crazed weasel. Not much different from any other supposedly-high-powered, straight-outta-b-school, dues-paying job like I-Banking or venture capital…
To be clear, there are good ways and bad ways to deliver solid Strategy Consulting services - and learning some of the tools that enable good delivery is very powerful. I regularly use my consulting tools in my current executive role and clearly differentiate my performance based on using them…
My experience at EDS was exactly like the OP, if not worse.
I jumped around a lot so I want to have at least a few years at one place. Plus I just got promoted so I want at least a solid year experience at my current level. Plus, I suppose, for now, it’s better to reign in Idiottown than to serve in some firm with a bunch of identical high-strung MBA peers. But long term, we’ll see.
As WordMan pointed out, most management consultants start in their 20s. Only the partners or “practice area specialists” are true industry experts. The rest start out grinding out Excel spreadsheets or Powerpoint “decks”. And firms usually burn through people pretty quick.
Like WordMan, I spent a year in a Big 4 firm as a part of a strategy consulting group. It is very schizophrenic since the firms often don’t specialize in that area and try to constantly shift you into developing hard service like SAP implementations.
Strike that. The other guy just quit.