This is a fascinating question, as I think many would argue that race in this play has actually been downplayed in some performances. Certainly the issue of race has changed greatly from the time this play was first written, and there is a definite sense in which, whilst the play hasn't changed, the issues that are most important to us about it has. Race does of course remain a vital component of the play, but its presentation has altered.
One production I have seen chose to focus on the issue of race through the lense of isolation rather than anything else. The play itself begins in Venice and then moves to Cyprus, and this move away from a busy city to a small island that is heavily defended leaves the main characters alone in a very claustraphobic setting that leaves them very isolated. All the characters suffer this isolation in one way or another, as Iago is shown to deliver lots of soliloquies and there are various examples of characters standing off by themselves, but the character who is most obviously isolated is Othello because of the colour of his skin and his physical stature. Iago is shown to be highly skillful in manipulating those around him, ensuring that Othello is more and more isolated so that he becomes his own worst enemy. Race as an issue never really entered the production except as a way in which Othello was isolated as a result from the mainstream community of which he was a part.