I teach yoga (BWY qualified via a long and repoutable course) and am pregnant. First trimester is a debate as someone said. Traditional schools of yoga such as ashtanga say no practice in first 14 weeks aside from pranayama, meditation but no asana work.
However, as other said this is a bit olde worlde and there are more relaxed approaches and if it feels good, go for it. However, NO twisting! Especially not deep as the placenta is forming/settling in trimester1 and the last thing you want to be doing iss twisting, squeezing and compressing the abdomen whilst this is happening. No reason why you can't do dog right now. As long as it feels good. Made me feel sick in tri 1 so I stopped and listened to my body,
Your teacher is right about no strong abs but I would, unless you have a very strong, consistent practice, suggest no strong abs in trimester 2 either. I'm 23 weeks now and no longer practice chaturanga/half plank etc or do anything with strong abs work but focus on legs and shoulders - areas you want to make strong right now. I work on the lower abs to keep the bump supported but not strong core - although there are various examples of very experienced yogis carrying on until right until birth with poses requiring strong abs such as headstand but these are the exception rather than the rule 
Obviously, in tri 2 I wouldn't have you on your belly at all and always lie on your left hand side. You can still be on your back but as you move through trimester 2 and into 3, you must not be on your back as the weight of the placenta can compress the vena cava interrupting the blood flow to both yourself and the baby. Savasana or final relaxation on your left side. In tri 3, you need to remember the relaxin is loosening up the ligaments and you can overstretch so many hip openers are off the list then as the last thing you want to do is destabilise the pelvis by over-stretching. In tri 3 - I'd maybe try and find an ante-natal class by then.
There is TONS you can still do but you do need to modify your practice a lot and I'm ensure you really do have a teacher who knows the principles. As someone else said there are specilist teachers trained for pregnancy. Other general teachers should know the basics to keep you safe but sadly the yoga teaching industry is not as regulated as it should be and there are teachers out there who've taken short cut routes in who really aren't perhaps up to speed as they should be.
Hope that's helpful in some way and enjoy yourself. I'd not survive without my daily yoga and it's such a nice way to keepo yourself well. 