Today, I had a co-worker ask me a question I hadn't really given much thought to. She asked me, "Do you think our loved ones still hear us after they're gone." Since I was busy, I told her, "I don't know," and went about my business. Recently having a grandmother I love greatly pass away, this question intrigued me, so I put some thought into it. I fully intended on talking to her about it later, but it never happened. That was a missed opportunity, and I don't want to miss an opportunity to answer this question. This is not an answer directed at her loved one, but is a response intended to cover the majority of those who would ask this question.
The Scripture cited at the beginning of this post is from a parable that Jesus was telling. In it, we see mentioned "Abraham's Bosom" (Heaven) and hell. It talks about two people who have died, each going to a different place. Since this is a parable, everything in it is not to be taken literally, but allegorically. From it we discern that, after death, there are only two places people could end up. Each one is freely chosen by the one who is sent there. Once there, there is no turning back. It is with this understanding that I would like to answer this question.
Hell is one of the two options. Those who go there have freely chosen to reject God and Jesus Christ. Remember that hell is a place of torment. Those there will never meet the most important Person to ever walk the face of the earth. There is enough evidence in nature for God that anyone who rejects Him has freely chosen to do so. Paul the apostle made this clear in Romans 1:20, and so have those who study natural theology. We can gain at least a basic understanding of Who God is by simply looking at His creation. The free choice to accept or reject this is up to the person.
Heaven is the second option. Remember that Heaven is a place of unspeakable joy, where we cannot even put into words the joy that we will see. Even if we could comprehend them, words could not express them. Those who go here accept the most important Person to ever walk the face of the earth, and will meet Him face to face. There is no greater joy for them than this. They have given up everything in order to find the One who made everything.
Now that we understand the two choices, we can say that the answer is, more probably than not, no. The reason is simple: those who have chosen hell experience no joy, including speaking to loved ones. Those in Heaven are already experiencing all of the joy they could possibly experience by being with the One for whom they were made. If it were possible, speaking to them would only diminish the joy they are experiencing. All the words we could say and all of the love that we can give are just a drop compared to the ocean of things they are experiencing in Heaven---things so great that no human language could adequately express even the smallest of these experiences.
If we truly love our departed loved ones, we will seek Jesus. For those of them who are in hell, I am sure that they would want you to find the joy that they missed. For those of them who are in Heaven, I am sure that they would not want you to miss the joy they are experiencing. So regardless, we are left with a choice: reject Jesus and miss the joys of Heaven, or accept Him and gain this joy that some of our loved ones are surely experiencing with Him.