The following table will show you what dice to roll to simulate the new set of d3, d5, d7, d14, d16 and d24 dice. (NOTE: to simulate these new dice shapes and have an equal chance rolling each possible result, it is important to avoid rolling more than one die, unless one of them is a "control" die, that adds a fixed amount to the result.)
d3 = roll 1d6, halve the result, add 0.5 to that if you rolled an odd number.
d3 = roll 1d4, if a "4" comes up, immediately roll again.
d5 = roll 1d10, halve the result, add 0.5 to that if you rolled an odd number.
d5 = roll 1d6, if a "6" comes up, immediately roll again.
d7 = roll 1d8, if a "8" comes up, immediately roll again. Do not use 2d4-1.
d14 = roll 1d20, re-roll anything above 14.
d14 = roll 1d8 and a 1d6 control die. Re-roll any 8s, and if 4-5-6 comes up on the d6, add 7 to your total.
d16 = roll 1d20, re-roll anything above 16.
d16 = roll 1d8 and a 1d6 control die. If 4-5-6 comes up on the d6, add 8 to your total.
d24 = roll 1d12 and a 1d6 control die. If 4-5-6 comes up on the d6, add 12 to your total.
d30 = roll 1d10 and a 1d6 control die. If 1 or 2 come up on 1d6, add nothing. If 3 or 4, add 10. If 5-6, add 20.