I just turned 59. An alarm has got me out of bed maybe 4 times my entire life. I’m one of those weird people that can tell myself what time I want to get up in the morning and I will get up at that time.
I can go to bed at 1am and if I have to get up at 5am no problem. Drives my wife crazy. 
And candles were expensive to buy and labor-intensive to make yourself. Most families would have avoided their use as much as possible.
In the early days of the Industrial Revolution, before cheap alarm clocks were available, workers generally lived quite close to the factories, which could use bells or, later, whistles to wake the workers up. Combined with fear of losing your job, that would do the trick.
Roosters and church bells.
I read once that you can alter your cows’ clocks as well as your own. They don’t HAVE to be milked at 0 Dark Thirty in the morning. You can slowly change it so you can milk them whenever YOU want to milk them.
Alarm cocks. 
I imagine a society where there’s no electricity and so activities are (generally) dictated by sunlight will mean you get to bed at a reasonable time, thus allowing you to get up at one as well. Along with no electricity comes no TV where you sit staring at it until 11:30 and then say “holy crap. It’s 11:30! We should go to bed.”
There also wasn’t the social stigma of an early bedtime so that, were you to go to bed at 8PM you wouldn’t have the need to post on Facebook “OMG, it’s 8PM and I’m tots in bed already! G’night y’all!” and then spend the next half an hour refreshing to see if anyone replied to your status.
On The Cosby Show once Cliff was talking about how his father used to brag about how he’d get up every morning really early with no rooster, no alarm clock, just “a sense of responsibility.”
Size matters?
I hardly ever get up with my alarm any more. Not responsibility - just stupid dog who tells us that it is time for her breakfast. She goes by the light.
This is also an old military trick for waking up for watch.
You hope.
Did the roosters wake up the church bell ringers who…Oh never mind…
Come to think of it, animals generally. In addition to food animals in agrarian settings, as pointed out by Chefguy and RealityChuck, the most common household pets generally obviate the need for mechanical contrivances[sup]1[/sup]. Anyone who’s ever lived with a dog or cat knows this.
[sup]1[/sup]Probably my favorite gag in the movie Babe is how the real Rooster resents and despises an alarm clock, which he terms a “Mechanical Rooster”.
Alarm Sundials - wadja think!
I haven’t had a regular job in years, and I’ve been living in a rural/self-sustaining way, and how I wake up have been pretty much described:
- Unless on very rare occasions, I am never sleep deprived. I go to bed when I feel tired, even it’s 9pm (in winter, more like 11pm in summer).
- I wake up with the sun and/or with the song birds.
- There is nothing time sensitive happening in the morning that I won’t be already woken up for anyway. (Exception is today, I have to drive my partner to the city to take a train, I put an alarm clock but I woke up an half hour before it anyway)
I only have goats and hens and they don’t care if I go visit them at 9am vs 4pm.
This gets less feasible the farther from the equator you live. At some latitudes there will be times of the year when you would not get more than a couple hours of sleep; at the extremes you’d be in bed and then awake for six months at a time.
If you lived in a big house with several family members, I imagine the first one up would get everyone up. There’s always one person in every family who can’t sleep past 5 am
A lot of good answers, but remember there was no electricity and sources of light (candles, lamps) where prohibitively expensive for most. This meant you went to bed when it got dark, and generally night time lasts for longer than a human can sleep. Also since there was no artificial lighting, this meant nothing could really be done except during the day, so you would have no need to wake up before the sun was out.
This, along with the thread subject reminded me of an old SD column; How come I wake up just before the alarm goes off?