There's an easier way, at least up to element 18 (argon).
If you just stick all the symbols for all the elements together it makes a big word:

HHeLiBeBCNOFNeNaMgAlSiPSClAr
pronounced
H-Heli-Beeb-Canoff-Nenarr-Magal-Sips-Clar.
Well, it works for me.