I get very horny during my period and the few days just before especially - could sometimes just pull any old man in off the street! I think it maybe something to do with your body recognising a viable egg is just about to get flushed away and is trying to insist you get inseminated before this happens!
However I'm not too keen on sex while on a period if there is nothing to stem the flow - not because I worry about a man getting blood on his cock - after all most are willing to get theirs covered in shit for anal sex and like to liberally spray us with foul tasting semen!, but because it messes sheets etc up. I once had sex just after my period had finished - or at least I thought so, but obviously the action brought me on again. It was hilarious as we were making love in the dark on his cream sofa - the lights went on and it looked like the Texas Chain Saw Massacre!
Following this I learnt a trick during my time working as an escort - many working girls don't want to loos a week's earnings and so use sponges which stem the flow. These are small round sponges about the 2 inches in diameter or you can cut them out from larger sponges. These work really well depending how heavy you are, and you really have to learn the knack of getting them in and out - they should be damp and covered in lube on insertion, squeezed together and pushed in with a finger, they should be removed immediately after sex or you run the risk of developing toxic shock syndrome, I always found staying calm, relaxed and often lying down on my back enabled me to get my finger in and manage to get in out - yep you certainly get to know your own body by doing it! and of course they have to be scrupulously clean (boilied before hand). When I used them none of my clients ever knew I was on my period - no mess, odour etc. Someone also mentioned caps which work equally well and are probably the more sterile option, although they are reused whereas the sponges are thrown away after each use. If you have any worries regarding them best consult your sexual health clinic.
Also for all you guys and girls who enjoy oral sex but aren't so keen during menstruation - try having the lady wear a tampon if she is able and then use a dental dam to cover her vulva and vagina - the sensation is the same as without. I always used them with clients for oral sex on me as I didn't fancy catching genital herpes, clamydia, gonnorhea, Hep B etc etc. You can usually get them from sexual health clinics, or if not you can order them online easily enough - I think britishcondoms.co.uk still do them.