Is there a chiyuv now days to go to the mikvah after relations?
This is not an obligation. Authorities (even from the time of the Gemara; see Berachos 22a) rule that the enactment of Ezra is not binding today, and it is permitted to daven and to study Torah without immersion (see Shulchan Aruch Ha-Rav, Orach Chaim 88:1).
However, this is not to say that there is no virtue in doing so. The Rambam writes that although there is no obligation of tevillas Ezra, he himself was never mevatel going to the Mikvah, and according to the Rif it remains proper to do so before prayer (see Rabbeinu Yonah, Berachos 13b, who writes that even if there is no obligation, a person’s tefillos are more readily accepted after immersion).
For somebody who does not go to the mikvah, there remains a virtue of washing with 9 kabin of water, and a shower is possibly sufficient for this. Please see here.