I don't know how your HRM is set up, but with mine, you can detach the strap from the part with the contacts and the transmitter. Periodically I'll throw the strap into the washer (front-loader) in a lingerie bag, usually with stuff like cycling gloves, and air dry it.

I wouldn't put the transmitter or the contacts in the washer, whether they're separate pieces or one piece, just because of all the twisting, even in a front-loader. They have to have some degree of water resistance to deal with sweat and rain, and some of them are used by triathletes in the water. When the contacts get cruddy (which happens more often with some designs than others, and also more often with some types of electrode gel than others, if you use any) - I take a toothbrush to it. Takes two minutes and gets it cleaner than machine washing would, anyway.