I’m going into the Oval Office to work something out, and can’t leave till the paperwork is done.
In the words of Homer Simpson, I have to shake the dew off the lilly.
“Voy a cagar”
Unless I’m around spanish speakers, in which case I just slink away quietly.
No. 1
Drain the main vein
Bleed the dragon
Crack china
No. 2
Ride the porcelain Honda
going for a dump
You gotta love 'em all. There is no phrase choicer than one which lets other people know that you’re going to move your bowels.
Release a harbor trout.
Pass some beavers through the lock
Build a bear
There’s these two Dutch comedians who pretty much based an entire 15 minute skit on #2 euphemisms. It cracked me up so bad I nearly fell off the couch watching it.
Now, bear in mind, we’re not as PC as the average North-American, but I assure you all examples below are harmless, and without ill will.
**I’m going to put Nelson Mandela on the train.
I’m going to knit a brown sweater.
I’m going to fax to Darmstadt. (All right, that one needs an explanation. Darmstadt is a German city, but the word darm in Dutch means “colon”, so “faxing to colon city”)
I’m going to rip open a bag of gardening soil.**
I guess you had to be there, and I guess understanding the intricacies of a very thick Rotterdam accent wouldn’t hurt either. But it was funny. 
My grandfather used to say, and now I say “I have to go visit the Pope”.
The euphanisms I hear amongst my friends are:
“Take the Browns to the Superbowl”
“Drop the Cosby’s off at the pool”
We use “Frank #2” around here. Especially if it’s one you know is about to get messy.
“drown a pommie”…(sorry brits)
Gays only: Download compressed log files.
Otherwise- lay a loaf
If diarrhea- the Hershey Squirts
A trip to the mens room - “I have to go consult with my attorney.”
“taking a dump”
to which a friend of mine would always respond, “why do they call it taking a dump when you a leaving it?” usually in a hilariously exagerated indian accent ala Apu of the Simpsons.
Another friend would say this when he needed to take care of #1: “As Pericles said to the Athenians, ‘Where may I drain the lizard?’” I don’t know why but this always struck me as hilarious.
I’ve also been known to say: Time to for a download/system dump.
stock the lake with brown trout
tap a kidney
going to the bog
nah that sucks “goin bog” much more better.
In school there were a few of us who would say “I’m going to look for Elvis” the usual response when you came back was “Did you find him?” “Nah, but I might have seen insert name of crappy popular singer of the day”
My wife tells our dog to “go do your business” when she lets him out into the back yard.
So when he goes back and eats his shit, I guess that’s what’s known as “the Businessman’s Lunch.”
My mother called what was in my kids diapers when they woke up:
“their morning constitution”
I hope she didn’t use that when I was that young.