Just like Multiplication table keep reading and repeating the procedure.
The best way to remember is, read first 10 elements and try to remember it and when you are done with first 10 elements move to next 10 elements. Now remember 20 elements in stretch.
when you are completely confident that you remembered the first 20 elements then only move to next 10 and then next 10 and follow the same procedure.
The ultimate way to remember will be read it in a rhythmic form and follow same process as mentioned above.
Practice makes man perfect, repeat the procedure everyday.
You can also use Periodic table poster and paste it on the wall. Read the elelment when you pass through the poster.
Learning both by vision and hearing makes it easy to remember and understand.
I tried it and now i remember 100 elelments.
Sanjeetmanna's idea is a really good method. You can also make flashcards, or use the website below to help you.