AA, AB, AD, AE, AG, AH, AI, AL, AM, AN, AR, AS, AT, AW, AX, AY
BA, BE, BI, BO, BY
DA, DE, DO
ED, EF, EH, EL, EM, EN, ER, ES, ET, EW, EX
FA, FE
GI, GO
HA, HE, HI, HM, HO
ID, IF, IN, IS, IT
JO
KA, KI
LA, LI, LO
MA, ME, MI, MM, MO, MU, MY
NA, NE, NO, NU
OD, OE, OF, OH, OI, OK, OM, ON, OP, OR, OS, OW, OX, OY
PA, PE, PI, PO
QI
RE
SH, SI, SO
TA, TE, TI, TO
UH, UM, UN, UP, US, UT
WE, WO
XI, XU
YA, YE, YO
ZA

Here's an easy way to memorize them. Separate them into vowel-consonant, consonant-vowel, vowel-vowel, and consonant-consonant categories:

vowel-consonant: A : BDGHLMNRSTWXY E : DFHLMNRSTWX I : DFNST O : DFHKMNPRSWXY U : HMNPST

consonant-vowel: BDFHLKMNPTYZ : A BDFHMNPRTWY : E BGHKLMPQSTX : I BDGHJLMNPSTWY : O MNX : U

vowel-vowel: AA AE AI OE OI

consonant-consonant: SH MY MM HM BY

There are only 5 each in the latter two categories, so just memorize them. For vowel-vowel, they start with A and O, and end with E and I, together with the first 2-letter word in the dictionary AA. For consonant-consonant, BY and MY are common words and SH, HM, MM are all sounds where each word starts with the letter the previous word ends with.

For the other two categories, we use mnemonics to remember which letters can precede or follow the vowel. Ignore the vowels, and the consonants in each phrase will yield legal Scrabble words.

vowel-consonant A : BIRTHDAYS MANGLE WAX E : WHITE SOX FARMLAND I : INFESTED O : SYMPHONY FAX WORKED U : PHANTOMS

consonant-vowel FUNNY KLEPTO ZOMBIE DEATH : A MET WHIPPED BOYFRIEND : E MAKE EQUAL SHAPE LIGHT BOX : I HELPS TWANG BANJO MELODY : O X-MEN : U