I will sound stupid, but I did not try to cycle my tank... I dont even know if it cycled or not... I just have my fish, 3 plants and gravel...I know that My ph is 7,4 in one tank and 7,6 in the other. But its about it...
If your filter has been running for a bit, it has at least started to cycle. Fishless cycling is popular, but it is something that will happen naturally even if you aren't trying.
If you are afraid to leave the filter on, I would put the filter media in the corner of the tank (maybe in a mesh bag?) and run an air stone near it. Just keeping it moist in a baggy won't be good enough to keep it alive, and if they bacteria in it dies it could cause a new cycle to start that could hurt your fish. The bacteria needs oxygen, so putting it where there is an air stone moving the water some should keep part of it alive at the very least.
Personally, I would go and buy a cheapy box filter (the kind that hooks up to an air pump) and stick the filter media inside it if it will fit.
I definitely would not just turn it off for a week... And if I did, I wouldn't expect my fish to be alive when I came home.
I have to leave for a week. I'm worried that my fishes could get stuck in the filter while I'm away. Would it me bad to turn off the filter for 7 days?
The tanks are Fluval Chi 5 g. They have real plants inside and a heater. I'm planning to let the fishes feast during that period. Could I get bad surprises at my return?
how many fish?
If its just one Betta in your 5g with plants you can kill the filter and your fish should be fine as for food try a feeding block you can get them at most pet stores and they make one just for bettas, so that way you dont end up with rotting food, i run lots of my 5.5g tanks with out filters and i just do water changes so just do a realy good water change when you get home.
as for your cycle you can try running your filter in a bucket with some tank water and some new water, use this water to do your water change s soon as you get home.
I'm not sure if you meant feast or fast. IIRC Betta's can go up to 9 days without food. So if you meant fast there won't be problem just give them a feeding before you leave and I would give them smaller portions more often when you get back and not try to "catch up".
Now if you meant feast you could definitely have problems as some (most) Betta's will eat even after they are full. This could lead to constipation, SBD, and other problems with water quality.
I would go to petco/petsmart and buy some sponge media such as fluval filter sponges and cut them so that they cover the intake. Like others said you probably have a cycle and don't realize it as it does happen naturally.