Can we hear God's - yes.
Do we hear like Moses or Elijah - often not, because we can experience the Holy Spirit differently.
God can speak to anyone, even a fallen murderer like Cain, but before Jesus made the Holy Spirit available to all believers, he would come and go and rest upon some from time to time.
Since Jesus' was resurrected and glorified the Holy Spirit actually comes to dwell in the believer and speak directly to our spirit, which then communicates through our soul to our mnd and human brain, which is why God's voice so often sounds like our own and speaks to us in the first person.
In Matthew 16:15-18, we read:
He said to them, "But who do you say that I am?"
Simon Peter replied, "You are the Christ, the Son of the living God."
And Jesus answered him, "Blessed are you, Simon Bar-Jonah! For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my church, and the gates of hell shall not prevail against it."
The 'Rock' upon which Jesus says he will build his church is the living principle of the believer's ability to hear God's voice directly through the Holy Spirit.
Of Moses and Elijah, Jesus had this to say in Matthew 11:11,
"Truly, I say to you, among those born of women there has arisen no one greater than John the Baptist. Yet the one who is least in the kingdom of heaven is greater than he."
John was the last great Old Covenant prophet, of whom Jesus says there has been no greater (including by implication, Moses and Elijah). Yet, even John would be lesser than the least believer because of the New Covenant believer's reconciled intimacy with God through the Holy Spirit, in a way not open to earlier generations.