There are no problems with the sentence "Due to his poor memory, Richard forgot about the appointment".
The other sentence, "Due to Richard's poor memory, he forgot about the appointment" contains a possessive antecedent (Richard's poor memory), which has offended some prescriptivists.
Wikipedia has a short entry on the possessive antecedent and why some object to it. The entire entry is reproduced below:
In English grammar, a pronoun has a possessive antecedent if its
antecedent (the noun that it refers to) appears in the possessive
case; for example, in the following sentence, Winston Churchill is a
possessive antecedent, serving as it does as the antecedent for the
- Winston Churchill's history shows him to have been a good writer.
In the 1960s, some usage guides started to reject the use of
possessive antecedents. These guides argue that a pronoun's antecedent
cannot be a noun in a possessive construct; in this case, they contend
that Winston Churchill, embedded as it is in the construct Winston
Churchill's, cannot serve as the antecedent for the pronoun him. The
basis for this contention is that a pronoun's antecedent must be a
noun, so that if Winston Churchill's is an adjective, then a pronoun
cannot refer back to it. This rule does not reflect ordinary English
usage, and it is commonly ignored (intentionally or otherwise) even by
those who have heard of it.
There is also a discussion of the topic by Professsor Pullum on Language Log, based on a controversy as to whether the following sentence from a question in a PSAT test contains a grammatical error or not:
Toni Morrison's genius enables her to create novels that arise from
and express the injustices African Americans have endured.
You can read more about the issue by doing a search on PAP or Possessive Antecedent Proscription.