Data Logging without Interfacing SD Card module
The simplest answer is to install a program that handles serial connection and can write everything into a logfile. On Linux, CuteCom is a program like that. On Windows, putty can handle serial (as well as SSH and telnet), and has logging.

The more advanced soulution is to write a program (or a simple script) that can do that. You could make it to do some other stuff as well (like configuring the Arduino side), and you can run it in the background without user interaction.