Best substitute wine glass
Trying to keep your body heat from the wine is only necessary for chilled wines (whites, for the most part).  For those served at room temperature (most reds), you actually want your body heat to warm the wine.  

I've been to plenty of places that use 'stemless wine glasses' (about the size of a red wine glass, maybe a little larger, but no stem underneath) for reds.  I've also been to places that just use mason jars.

For whites, if you're at a dinner table, it's not going to matter as much, as you can put your glass down.  The problem comes when you're standing around with your glass.  You can get around the problems by making sure that the wine is well chilled, and don't serve as much per glass (so they don't hold it so long that it warms up).  Handled glasses can help, but I wouldn't go for a large beer mug like Cos recommended -- I'd use a smaller mug for coffee or tea.

One exception would be your bubbly wines.  You want something tall and narrow for those.

None of these have the characteristic wine glass shape, with the smaller opening at the top, which is going to help concentrate the aromas, so if you have any glasses that are shaped like that, consider using those.  And you don't want to fill any glass of wine more than half full, so that you have space for the vapors to collect.