I'm back with more research. In the original Greek, the word used there was "adialeiptōs," which is typically translated as "without ceasing." Looking up its definition, I got "without intermission, incessantly, without ceasing." The word was also used as part of the following verses (KJV because it's a more direct translation):

For God is my witness, whom I serve with my spirit in the gospel of his Son, that without ceasing I make mention of you always in my prayers (Romans 1:9)

We give thanks to God always for you all, making mention of you in our prayers; remembering without ceasing your work of faith, and labour of love, and patience of hope in our Lord Jesus Christ, in the sight of God and our Father (1 Thessalonians 1:2)

For this cause also thank we God without ceasing, because, when ye received the word of God which ye heard of us, ye received it not as the word of men, but as it is in truth, the word of God, which effectually worketh also in you that believe. (1 Thessalonians 2:13)

And in adjective form:

that I have great heaviness and continual sorrow in my heart. (Romans 9:2)

I thank God, whom I serve from my forefathers with pure conscience, that without ceasing I have remembrance of thee in my prayers night and day (2 Timothy 1:3)

The most common interpretation among theologians seems to be the "keep the comms open" kind of thing, but it looks like another valid interpretation, especially given that last verse, is more like "don't take any days off from prayer." Nor periods of the day. In other words, be praying a lot, and never pause that practice.