For most sets of shapes a periodic tiling is possible, but by no means guaranteed.
For example, consider rectangles with sides of 1 and 3 units: they can cover the plane periodically (e.g. in a simple rectangular grid), but also aperiodically, because you can form a square grid of square 3 by 3 units "metatiles", each encoding one bit of information in the vertical or horizontal orientation of the narrow rectangles; then it's easy to break symmetry by orienting metatiles so that for all integers m and n some metatile differs from the metatile m rows and n columns away, so the period cannot be m rows and n columns.
For example, consider rectangles with sides of 1 and 3 units: they can cover the plane periodically (e.g. in a simple rectangular grid), but also aperiodically, because you can form a square grid of square 3 by 3 units "metatiles", each encoding one bit of information in the vertical or horizontal orientation of the narrow rectangles; then it's easy to break symmetry by orienting metatiles so that for all integers m and n some metatile differs from the metatile m rows and n columns away, so the period cannot be m rows and n columns.