This expands CodesInChaos's comment into an answer.
Forward Secrecy (that is, maintaining confidentiality of messages enciphered before compromise of the long term key) can be achieved in a protocol using a public-key signature scheme with a long-term public key, and a public-key encryption scheme with a per-session key; but in the case of RSA signature and encryption, that's inefficient, thus unusual.
As an example: Bob has a long-term RSA key pair $(Mpub_B,Mpriv_B)$ used for signature, with $Mpub_B$ trusted by Alice (perhaps by way of some certificate). In order for Alice to send a confidential message to Bob:
- draws a 256-bit random $R$
- sends $R$ to Bob
- generates a new RSA key pair $(Tpub_B,Tpriv_B)$ used for encryption,
- RSA-signs the (hash of the) message $R\|Tpub_B$ using $Mpriv_B$ giving signature $S$
- sends $Tpub_B\|S$ to Alice
- gets $Tpub_B$ and $S$
- verifies that $S$ is a valid signature with respect to $Mpriv_B$ for $R\|Tpub_B$, where $R$ is from the recent first step
- generates a random symmetric session key $K$
- RSA-enciphers $K$ using $Tpub_B$ yielding $X$
- enciphers the plaintext message $M$ using key $K$ by a symmetric algorithm (say, AES-CTR will implicit zero IV) yielding ciphertext $C$
- forgets $K$
- sends $X\|C$ to Bob
- gets $X$ and $C$
- RSA-deciphers $X$ using $Tpriv_B$ yielding $K$
- forgets $Tpriv_B$
- deciphers ciphertext $C$ with key $K$ yielding plaintext message $M$
- forgets $K$.
$K$ allows $M$ to be large, when RSA encryption only directly allows short messages. $R$ protects against replay of an earlier $Tpub_B$.
The scheme is inefficient because the generation of a new RSA key pair is relatively expensive (and normally rare, thus not optimized for speed). That's a good reason why (EC)DH is most used in practice.
It is possible to send several messages using the same $K$, or/and reuse $(Tpub_B,Tpriv_B)$ across multiple sessions, improving performance. But Forward Secrecy triggers only when $K$ and $Tpriv_B$ are forgotten, and $R$ is no longer accepted.
Note: the scheme provides confidentiality, but not integrity or proof of origin; that can be added.