So stick to RS232 will be alright?
Maybe, maybe not. It isn't clear whether you're talking about RS232, or logic-level asynchronous serial. The timings are all the same, but the voltages are very different. If it's connected directly to the pins of the UART chip, it's likely to be logic-level. If it's on a connector, there's a chance it's RS232 (but it might not be).
There's no standard name for the logic-level version (this is just what I call it), and people often call both RS232. Before you go plugging displays into it, put a multimeter on the data line. If it's negative when idle, you've got RS232 (-12V is typical). If it's 3V or 5V or somewhere around there, it's logic-level.
Newhaven have some displays that say "RS232 (TTL)", meaning logic-level. They would get very unhappy with real RS232. lcd-module.com have some that appear to take both (on different pins).