I ferment in carboys, so pouring through a strainer doesn't work straight to the carboy. This is how I get around it.
During brew day, I fill my bottling bucket with sanitizer water. I soak all my post-boil stuff in there during the entire process (including a strainer).
While chilling the wort, I fill a different pot (the one I use for heating sparge water) with the water from the bottling bucket and transfer everything to it. Dump the remaining sanitized water on your bushes, plants, etc. (living in Texas, I try to reuse / conserve water where I can and haven't noticed any ill effects on the bushes, trees, or grass).
Place the strainer over the bottling bucket and pour the chilled wort through it catching the hot-break. Pour the wort from the bucket to the carboy through a big funnel. This acts as my oxygenation step and trub filter at once.
That was a really long description of a simple process.