I had the same prob. my little gal is now 5 and a half months old and she still does not sleep on her back.
The way i see it is its up to you, she is your daughter. I know that people tell you otherwise, and it did take me awhile to get use to the idea of sleeping my daughter on her tummy, As we all know SIDS and what not.
Also, when we slept her on her back, she would wake every hour or so in the night, then when switched to tummy sleeps like an angel. 6 30 till 6.30 its great smile
If you are worried there are things that you can get to put your mind at ease, such as mats that alarm when the baby stops breathing etc. But if you do get something like that (which might I add are quite expensive) I would recommend you take a first aid course, therefore you will know what to do if your baby stopped breathing.
I spoke to my child and youth health nurse about this issue alot and she told me to try and settle her on her back, but if not well do what you feel comfortable doing.
Now my gal has lovely neck control, loves playing on her tummy (which I know many babies who hate it) and its no longer an issue, because of the neck control. I did find that she developed things like neck control much earlier because she slept on her tummy.
the choice is entirely up to you.