This happens to me, too, and I'm not using a Janome. The problem is that the needle is not quite (or just barely) on the fabric and the upper and lower thread make a knot that catches in the feed dogs. Usually the fabric edge is held together by the knot, thus the appearance of 'eating' fabric. If you have long lengths of bobbin and needle thread when you start to sew, it is more likely to happen. Experiment to see the minimum thread length you need to avoid, or at least minimize, the problem.
One of my machines has a thread cutting button that can be used to snip the threads when I'm done sewing a piece or a chain of pieces. I am amazed at how little thread is actually required to begin sewing. On my machine, the snip leaves only about 1.5 inches of thread. If your Janome has a thread cutting button, experiment!
Solutions for eating the fabric have been mentioned in earlier responses - use a leader scrap to start sewing, smaller needle, hold the threads as you start to sew, and, as you guessed from the paragraphs above, make sure you don't have excessive bobbin and needle thread lengths when you start to sew.
Good luck with your new machine!