I believe Jesus intended that we should remember Him by regularly sharing meals together (call them eucharistic meals if you like). This would be the perfectly logical way to live out His message about living in the Kingdom of God, as kingdom-people. Jesus teachings about the Kingdom focussed on living in community, and at the last supper He used the language of Jeremiah to announce that the age of the new covenant was beginning, when God would deal with His covenant-people in a new way. He tied together "remembering" with looking forward, and spoke of the messianic banquet in the age to come when His new-covenant community would come together in a climactic meal.
The language of kingdom and covenants directs us to think in terms of community. Paul picks it up by using the Greek word used in the Septuagint for the community of God's people when they assembled together - ekklesia. What Jesus announced in His Gospel of the Kingdom Paul developed with his practical advice about assembling together as God's people - His church/ekklesia - and living in community.
However, if we were to remember Jesus in the way I believe He intended there would be implications for the way we "do church". Here are just a few suggestions:
1. Our focus would change from the 'assembly hall' to the 'assembled people'. God wants to dwell among His people, not in bricks-and-mortar buildings. Many of the people we are called to reach will never go near our buildings (which, by the way, sit empty most of the week). We must take Christ to the marketplace and meet people where they are, not where we are. We must meet people with the Gospel at their point of need. Formal religious services might be nice (especially for Christians), but for most people they are irrelevant. If our only interaction with other people where Christ is mentioned is inside a 'church' (or ecclesial hall), and in a structured setting, then it is artificial.
2. The emphasis would change from platforms to people. Too many of our spiritual 'conversations' are really monologues. Some people only share their religious experiences from the safety of a platform or pulpit, during an exhortation or sermon. We get used to the idea of people expounding the Bible from a position of authority ('the platform'), rather than to thrashing out the implications for our lives in real-life settings. In some ecclesias the selection of platform speakers is carefully controlled to ensure that only the 'official' view is ever taught, and the ecclesia is poorer for not hearing what God has been teaching the members in the pews through their daily encounters with Him.
3. The focus would change from membership to community. It's possible to be an active member of an ecclesia and not ever deal with other members in community. Someone can attend all the meetings, be active on committees, teach Sunday School, and be a speaker, without ever really getting to know other members of the ecclesia. What do the other members do for their living? Do you know their families? How do they relax? What are their interests? Do you know anything about their struggles, their accomplishments, their spiritual progress? Do you really know them at all? Do they really know you?
Take everyone out of their ecclesial halls and churches and put them in homes, around meal tables. Take them out of their suits and 'Sunday best' and put them in comfortable clothes. Take away their speaking notes and the commentaries in their wide-margin Bibles. Now let them interact with each other. Let them talk about their week. Ask them about what God has done for them. Let the conversations flow naturally and unstructured. Do this week after week and watch relationships develop and friendships form. Watch how community happens. And see how miracles happen when Jesus is remembered by breaking bread and drinking wine as part of a meal together, in the way He intended.