Yes, for just monitoring if _something_ is going on, you can probe the RXD/TXD lines with the DMM in ACV mode - provided, the meter has sufficient bandwidth, which would be a problem with cheap DMMs.
One more thing - do you try to interface the MCU directly to the COM port of a PC? I just want to make sure you are aware that the UART interface on the MCU is TTL, whereas the COM port on a PC is RS232. Even though the standard COM ports on a PC usually are TTL compatible to a certain degree, the voltage level will probably be too high for a MCU, especially if it runs on less than 5V rails. The usual solution would be an RS232 transceiver like MAX232 or one if its low voltage derivatives, but if you count on the TTL compatibility of the COM port, you would need at least some protection circuit to limit the voltage on the MCU port pins. Also serial ports on extension cards and USB to RS232 adaptors tend to _not_ have the before mentioned TTL compatibility.