Greek is the language the writers used to write all NT scripture. The above scripture in 1 Thess. is the only scripture about prayer that is interpreted as meaning to constantly be in a state of connection with God. If you think about what the writers were doing, they knew that God was using them to write down not only their experiences, but also what God was inspiring them to write, so they had to always be in a constant state of openness to God's Spirit...and then always be ready to write it down as well!
Therefore when they endeavored to write down what God was sharing with them about this unceasing kind of prayer, they used their own experiences...which was; never being able to take breaks without having paper with them! This in turn tells us that we are to always be in a communing and ready state with our Lord so when He wants to speak to us or use us in some way we will be ready. So! No paperless breaks for us!
Does this mean to be in a constant state of anxiety? Does this mean to be hyper-aware? No. Well, we'll probably start out this way...but this is because we're not sure how to do it yet. But it is supposed to become a natural thing...in fact it is what God wants for every spiritual practice - to become like second skin, like it was for Adam and Eve in the garden. (But of course at that time they didn't have to practice, they already had 'it' before their disobedience cut us off from it)
I would liken it to when you have a baby. At the beginning you are hyper-aware and even when she isn't crying you think she is. So every little noise you go running in there to check. Then as you practice, and as you continue to be open to her noises, you get to know what certain sounds mean and then you begin to relax and things become more natural. In fact every relationship is very similar. As you get to know each person you get more and more relaxed because you know how they sound and how they are unique in their own way...it is the very same with God. So, it takes some time and patience.
Learning this is a wonderful place to be in your life and walk, as it is what ushers in learning to hear God's still small voice much better, as well as being able to be in step with the Holy Spirit...which of course is what we all want so much! (Please view above picture...click on it...hope it helps!)
Subsequently I believe prayer means muchmore than is normally taught. Since I was young prayer meant to bow my head, close my eyes, fold my hands and pray aloud...but, as we know, folding hands and etc. are not biblical. However, since working with kids for so long, now I know where this started! Ha! You and I also know that God can hear us just fine even when we are not speaking the words aloud...even though there will be times one can do this, as in one's prayer closet.
Then as a young Christian I heard teachings that vehemently instructed that we had to incorporate daily devotions into our lives! The example that was given once was when the Israelites were provided with manna everyday they were instructed to not keep any extra for the next day because when the next day arrived the manna left over was wormy. So, it was used as a metaphor to say that the devotions of one day will not last till the next day!
However, when you and I learn more and more to allow our hearts to be open to God and have this kind of close relationship with Him...then because we are always in a peaceful, spirit of prayer, we don't have to worry about 'not missing prayer time!' So the prayer times you do have in your prayer closet, which may be at different times and needn't be planned, will be for those times when God wants you to be alone for studying His written word...praying about something specifically...worship...or whatever...but it will be what you and God wants it to be.