If you are doing a first person recreation of a medieval person, you should try to avoid obvious modernisms. It will improve the experience of those around you, and you may enjoy it more yourself. A truly accurate recreation of premodern speech isn’t necessarily an achievable or even desirable goal. If you are portraying an 11th c. Englishman faithfully, hardly anyone will be able to understand you, even if you could pull it off.
Avoid modern subjects. Here are some alternatives:
Fifty men at arms defeat a hundred. Which would you rather be, the worst of the fifty or the best of the hundred?
At a tournament "Which is to be more highly prized: the one who loses two horses or three in one day while attacking or defending quite openly, or the one who keeps his horse very close the whole day and endures and bears well the pulls and blows and everything that comes his way?"
Knights say they perform deeds of arms for the honor of their ladies. Is that so, or do they really do it for their own honor?
A man's wife is under an enchantment: she will be hideous during the day (when others can see her) but beautiful at night (when he sleeps with her.) Or the reverse can be true, but the man must choose one or the other, forever. Which should he choose?
When Adam delved and Eve span, who was then a gentleman? That is, since we are all equally descended from that pair, how can nobles claim that they are worthier than commoners?
There are many styles of music, vocal and instrumental. Which is best?
What is the greatest adornment of the mind, nobility of arms or letters?
Is it better for a man to be brave or wise?
Who was the wickedest woman in the Bible?
More examples of possible conversation topics can be found in Castiglione’s The Book of the Courtier, from which many of the preceeding are cribbed.
Avoid modern slang and contractions.
Avoid false archaicism. Medieval speech didn’t sound archaic to medieval people, it sounded contemporary, because it was.
Some in the Society for Creative Anachronism use medievalish terms when speaking about modern things: dragon for automobile and farspeaker for telephone. Others find these terms jarring, and there are usually better alternatives, like wagon or message. I prefer Head Cook to Feastocrat and Porter to Troll.
Use thee and thou correctly or not at all. The second choice is much easier.
Some people have the talent of picking up the patterns of speech of another period by osmosis. If they read enough Chaucer or Froissart in translation, they are able to convey some sense of the flavor and construction of speech from that time. If you’re one of them, great. If not, don’t worry about it. It doesn’t hurt to give the technique a try. If you’re interested in an era, read some good books written during the period. At worst, you’ll have read some good books and gained a better understanding of your period of interest.