First off, whoever told you that you can't wash your hair with microbraids was wrong.
Dead wrong.
Are your micros human hair? Synthetic? Are they tyzillion braids? If they're human hair, the best way to wash them is with diluted non-conditioning shampoo (three parts water to one part shampoo). Place the shampoo in a spray bottle and spray it over your head, making sure to get down into your scalp. Massage your scalp with the
pads of your fingers, not your nails. Be very, very gentle. Then as you work up a lather, you can squeeze the lather down the length of your braids. Rinse your hair, and if necessary, repeat. Follow up with a leave-in conditioner (Like Always 911, or Creme of Nature Herba Rich Leave-In Remoisturizing Treatment).
Do not use typical creme conditioners as they'll make the braids slip out.
If your micros are synthetic, you would wash the same way, but it might be okay to use a gentle conditioning shampoo (like Creme of Nature in the red and yellow bottle). The only exception to this is if your braids are made of micro-Yaki extension fiber, then they shouldn't be washed with water at all.
Another excellent product is Organic Root Stimulator Herbal Cleanse. It's latherless herbal shampoo that you put on your hair and scalp and remove with a dry towel. I've used it on my locs and my hair was just as clean as it was if I'd shampooed. This is the perfect solution for micro-Yaki braids, because your hair doesn't get wet and tangled.
Follow up your shampoo with a light hair oil (I like Khoret Amen Oil by Carol's Daughter, but Kemi Oyl or Hot Six Oil work ust as well). If you want your hair to dry neatly, tie it up with a scarf and sit under a hooded dryer, or let them air dry - but the scarf will keep the fly-aways under control. At night, sleep with your hair tied in a scarf just like you would if your hair was relaxed. This will minimize the friction between the braided hair and your natural hair, and will protect your hair further.
Understand that using human hair for braids means that the braids won't stay in as long as they would if you'd used synthetic. You'll probably end up having to have them redone in a month's time -- two months if you're extremely careful.
If the braids are coming out already (and you just had them done today) march right back down to the braider and demand that they put them back in. Braiding shouldn't hurt, so speak up if they braid them too tight. You don't want to end up with traction alopecia. If the braids that are coming out are right along your hairline (front or back) you may just have to use a little bit of gel to keep that hair under control. You don't want to put too much stress on your hairline if you want to keep it intact.
For more good ideas on how to take care of braided hair, pick up a copy of Tulani Kinard's "No Lye" at your local bookstore or at Amazon.com. She gives detailed information on how to take care of braided styles.
Good luck, Soror!