I am often asked this question by sincere Christians. Their inquiry stems from their personal interactions with the Holy Scriptures where, at least for the Israelites, ongoing Shabbat observance seems to be presumed. However, usually Christians do not ask the Jewish Shabbat question (How must I keep Shabbat?) but rather they ask the Christian Shabbat question (On which day must I worship?).
These concerns are caused by the western mindset that prizes the corporate (joined) worship that occurs for most Christians one day per week, usually on Sundays. Coming out of this mindset, a Christian logically asks on which day he or she must worship to be pleasing to God in Christ: Sunday or Saturday? For a Jew, however, even for the one steeped in western culture, this is never the question. For the Jew, “corporate” worship is an everyday practice and not a once-per-week occurrence. A Jew prays a lengthy set of prayers 3 times per day, often in the company of other Jews. This is why observant Jews wouldn't drive to a place of worship 15-40 minutes away (as most Christians do), but would only attend something local. Not only must the synagogue be reached by foot on Shabbat (when driving is off limits), but it also must be very near for everyday community functions.
On Shabbat, Jews engage in some worship activities which differ from regular daily worship, but most prominently, they cease from the creative life in which they engage during the rest of the week. (For example, a Jew can read but must not write on Shabbat, charging his or her creative strengths to be unleashed during the six days of work). This central Jewish idea of the once-per week physical, emotional, and psychological rest is almost completely absent in Christian practice, where emphasis is placed on worship.
Committed and involved Christians often come out of Sunday feeling physically tired from one or two services and possibly even a church related function. Obviously, this description is not at all universal and does not apply to all Christian practice, but I think it is a fair characterization of the general tendency in Christian communities. The words of Jesus that, "a man is not made for the Sabbath, but the Sabbath is made for a man" (Mark 2:27) would then make perfect sense in their own Jewish context. (For my thoughts regarding Gentile Christian observance of the Feasts of the Lord, please, click here).
Humans, for thousands of years, divided times and seasons differently. The week as a period of time did not always exist, and a week did not always contain seven days. The seven-day week was one of the great gifts that was established by the God of Israel and given by the divine-human culture of Israel to the entire world. That is to say, whether the Christian worships corporately with others on the first day of the week (Sunday) or on the seventh (Saturday), he or she already worships in accordance with the Jewish seven-days-per-week cycle, with or without recognizing it.
Obviously, there is a lot more to this discussion, but this is where any responsible thinking should begin.